Algebraic Markov Decision Processes

In this paper, we provide an algebraic approach to Markov Decision Processes (MDPs), which allows a unified treatment of MDPs and includes many existing models (quantitative or qualitative) as particular cases. In algebraic MDPs, rewards are expressed in a semiring structure, uncertainty is represented by a decomposable plausibility measure valued on a second semiring structure, and preferences over policies are represented by Generalized Expected Utility. We recast the problem of finding an optimal policy at a finite horizon as an algebraic path problem in a decision rule graph where arcs are valued by functions, which justifies the use of the Jacobi algorithm to solve algebraic Bellman equations. In order to show the potential of this general approach, we exhibit new variations of MDPs, admitting complete or partial preference structures, as well as probabilistic or possibilistic representation of uncertainty.